We compare several classical embedded pairs with the newly developped SSPERK embedded pairs. From the literature, we have the following methods:
RKF23, Ceschino24, RKF23b, Fehlberg12, Fehlberg12b, Bogackishampine, Merson45, Zonneveld43, Fehlberg45, and Dormandprince54.
The methods RKF23b [1]} and RKF23 [1]}: the solution \(u\)  advanced is second order with a third order embedded solution \(\hat{u}\)  used for error estimation.
Ceschino24 [1]}: \(u\)  is of second order while the embedded solution is fourth order accurate.
Fehlberg12 [1]}: the solution advanced is first order accurate, while the embedded weight used for error approximation is of second order;
Fehlberg12b [1]}: the embedded weight is second-order accurate, the method advanced numerically gives a slightly higher order than 1;
Bogackishampine [6]} is of order three with four stages with the First Same As Last (FSAL) property, so that it uses approximately three function evaluations per step (although we do not exploit this property in this study). This method is implemented in the ODE23 function in MATLAB.
Merson45 [1]}: often implemented in NAG softwares, \(\hat{u}_1\)  is of order 5 for linear equations with constant coefficients; for nonlinear problems it is of order 3.
Zonneveld43 [1]}: fourth-order accurate weight used to advance the solution while a third-order embedded weight is used for error estimation.
Fehlberg45 [1]} is a method of order four with an error estimator of order five.
Dormandprince54 [1]}, [11]} uses six function evaluations to calculate fourth- and fifth-order accurate solutions and is currently the default method in ODE45 solver.
